Methods and systems for computing a size distribution of small particles

ABSTRACT

Methods, systems and computer readable media for computing small particle size distributions. A reference matrix of pre-computed reference vectors is provided, with each reference vector representing a discrete particle size or particle size range of a particle size distribution of particles contained in a dilute colloid. A measurement vector of measured extinction values of a sample dilute colloid is provided, wherein the measured extinction values have been measured by spectrophotometric measurement at the discrete wavelengths. Size distribution and concentrations of particles in the sample dilute colloid are determined using linear equations, the reference matrix and the reference vector.

CROSS-REFERENCE

The application claims the benefit of U.S. Provisional Application No.61/068,101, filed Mar. 3, 2008 and U.S. Provisional Application No.61/068,098, filed Mar. 3, 2008, both of which are hereby incorporatedherein, in their entireties, by reference thereto and to which weapplications we claim priority under 35 U.S.C. Section 119.

BACKGROUND OF THE INVENTION

Many industrial processes involve the manufacture of particles and theproperties of the item of manufacture (pharmaceuticals, paint, food,chemicals, etc) depend heavily on the size of the particles used.Oftentimes these processes involve mixtures of particles, such as amixture of particles of one material having more than one particle sizeor a mixture of particles of more than one material wherein the mixtureincludes more than one particle size. However, current particle sizingtechniques do not generally distinguish between the different sizes ofparticles in a given particle size distribution and report a “universal”particle size distribution for all particles present in a sample. If afeature is observed, for example a peak due to fine particles, it is notknown which constituent of a mixture contributes to that particularparticle size.

Some current techniques for measuring particle size distributions forsmall particles, such as Dynamic Light Scattering, rely upon theBrownian motion of the particles to derive estimates of their sizemeasurements. However, computations of size estimates can take severalminutes and these techniques are therefore not ideal for onlineprocesses. Further, when the particles are not monodisperse (i.e., allof substantially one size), signals measured from movements of thelarger particles can tend to blur signals measured from movements of thesmaller particles to the extent that the smaller particles are notproperly measured, or not even detected at all.

U.S. Pat. No. 5,121,629 discloses a method for measuring particle sizedistribution and concentration based on directing ultrasonic wavesthrough a suspension of particles in a suspending medium. Sizedistribution and concentration calculations include fitting twolognormal distributions to the measurements, based on the assumptionthat the particle size distribution is the sum of two lognormaldistributions. There is no basis for this assumption and it sometimesleads to incorrect solutions. The Powell Discriminator described canerroneously lead to a local minimum solution that is not the overallglobal minimum solution and is therefore the wrong solution. Also, thesecalculations are not fast, taking on the order of thirty minutes tocalculate particle size distribution and concentration values for asingle measurement.

U.S. Pat. No. 7,257,518 discloses a method of calculating particle sizedistributions and concentrations of particles that are denselyconcentrated, so that multiple scattering effects must be accounted for.This method relies upon nonlinear methods of estimating the particlesize distribution and concentrations and can take a considerable amountof time to calculate measurement estimates.

Some existing particle size distribution estimation methods provideplots of estimates that are unacceptably noisy (e.g., spiky). There is aneed for improved methods for providing estimations that are smoothedand therefore provide more definite values for a distribution and valuesthat can be more readily read and ascertained by a user.

There is a continuing need for fast and accurate methods of measuringsize distribution of small particles, particularly for use in onlineapplications, for real time or near-real time calculations ofmeasurements during performance of a process where small particles areemployed. Even for offline applications, it would be desirable toprovide faster, accurate methods of measuring particle sizedistributions.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a schematic diagram of a spectrophotometer that may be used tomeasure scattering spectra of particles by spectrophotometry.

FIG. 2 is a flow diagram of a method for computing small particle sizedistributions according to an embodiment of the present invention.

FIG. 3 shows the attenuation spectra of dilute colloids of polystyreneparticles having sizes of about 100 nm and 5 μm at a concentration ofabout 0.1 volume percent, calculated over a wavelength range of about190 nm to about 1100 nanometers.

FIG. 4 illustrates a plot of the least squares error values againstvarious concentration vectors c.

FIG. 5A shows a particle size distribution calculated from a measurementvector and reference matrix according to the present invention, whenabout 1% noise was present in the system.

FIG. 5B shows the particle size distribution calculated from the samemeasurement vector and reference matrix used to calculate the particlesize distribution of FIG. 5A, but where much less noise (substantiallynone) was present in the system.

FIG. 6 is a particle size distribution plot illustrate large particlenoise artifact.

FIG. 7 illustrates changes in fit error relative to upper particle sizebound limit.

FIG. 8 shows the particle size distribution of FIG. 6, after postprocessing to remove large particle size artifact.

FIG. 9A illustrates forward scattering artifact in an attenuation plot.

FIG. 9B illustrates a plot having been corrected for forward scatteringartifact by applying a forward scattering correction factor. FIG. 9Balso shows the plot of FIG. 9A for comparison purposes.

FIG. 10A is a flow diagram illustrating post processing methods that canbe performed after computing a particle size distribution of a sampledilute colloid according to the present invention.

FIG. 10B is an embodiment of a subroutine that can be practiced in theprocess of FIG. 10A

FIG. 10C is an alternative embodiment of a subroutine that can bepracticed alternatively to that of FIG. 10B.

FIG. 11A is a plot of the scattering spectrum (attenuation spectrum)from 1 μm polystyrene particles prepared by Dow Chemical at aconcentration of about 0.1% by volume in water.

FIG. 11B is a plot of an objective function (equation (14)) as afunction of wavelength and real particle refractive index.

FIG. 11C illustrates a plot of refractive index results versuswavelength, as calculated using a point-by-point real refractive indexsolver application.

FIG. 11D shows plots of refractive index results versus wavelength, ascalculated using a point-by-point real refractive index solverapplication, for 1 μm polystyrene particles and for 300 nm polystyreneparticles.

FIG. 12A shows a plot of refractive index results versus wavelength, ascalculated using a Sellmeier real refractive index solver application,for 3 μm polystyrene particles.

FIG. 12B is a fit merit plot for the results shown in FIG. 12A.

FIGS. 13A and 13B show real and imaginary refractive index plots,respectively, for polystyrene particles.

FIG. 14 illustrates a typical computer system in accordance with anembodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

Before the present methods, systems and computer readable media aredescribed, it is to be understood that the terminology used herein isfor the purpose of describing particular embodiments only, and is notintended to be limiting, since the scope of the present invention willbe limited only by the appended claims.

Where a range of values is provided, it is understood that eachintervening value, to the tenth of the unit of the lower limit unlessthe context clearly dictates otherwise, between the upper and lowerlimits of that range is also specifically disclosed. Each smaller rangebetween any stated value or intervening value in a stated range and anyother stated or intervening value in that stated range is encompassedwithin the invention. The upper and lower limits of these smaller rangesmay independently be included or excluded in the range, and each rangewhere either, neither or both limits are included in the smaller rangesis also encompassed within the invention, subject to any specificallyexcluded limit in the stated range. Where the stated range includes oneor both of the limits, ranges excluding either or both of those includedlimits are also included in the invention.

Unless defined otherwise, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art to which this invention belongs. Although any methods andmaterials similar or equivalent to those described herein can be used inthe practice or testing of the present invention, the preferred methodsand materials are now described. All publications mentioned herein areincorporated herein by reference to disclose and describe the methodsand/or materials in connection with which the publications are cited.

It must be noted that as used herein and in the appended claims, thesingular forms “a”, “an”, and “the” include plural referents unless thecontext clearly dictates otherwise. Thus, for example, reference to “avector” includes a plurality of such vectors and reference to “theparticle size” includes reference to one or more particle sizes andequivalents thereof known to those skilled in the art, and so forth.

The publications discussed herein are provided solely for theirdisclosure prior to the filing date of the present application. Nothingherein is to be construed as an admission that the present invention isnot entitled to antedate such publication by virtue of prior invention.Further, the dates of publication provided may be different from theactual publication dates which may need to be independently confirmed.

DEFINITIONS

The term “monodisperse” refers to a sample containing particles whereinall of the particles contained are of substantially the same size.

The term “polydisperse” refers to a sample containing particles thatinclude a distribution of various particle sizes.

A “dispersion” refers to solid particles in a liquid.

An “aerosol” refers to solid and/or liquid particles in a gas.

An “emulsion” refers to liquid droplets in another liquid.

A “colloid” refers to particles in any phase in any fluid.

A “dilute” colloid refers to a colloid having a volume percentage ofparticles that is less than a volume percentage effective to producesignificant multiple scattering when the attenuation spectrum of theparticles is measured. Accordingly, multiple scattering effects do notneed to be accounted for when measuring the attenuation spectra of adilute colloid. In at least one embodiment, a dilute colloid is definedas one having particles in an amount less than about 1% by volume.

“Flowing” means that there is relative motion between particles and theinterrogating light beam, over and above relative motion due to Brownianmotion.

“Particle size” measurements refer to the diameter or largest crosssectional dimension of a particle. Particles are assumed to be roughlyspherical in cross-section. In practice, the term “particle size” refersto a small range of particle sizes.

A “particle size distribution” (occasionally abbreviated as “PSD”)refers to a characterization of the various particle sizes that arepresent in a colloid, and the relative concentrations of each. Theparticle size distribution can be represented by a histogram. Each binin the histogram refers to a small range of particle sizes displayed asthe x-axis of a plot. The number of particles in each size bin isdisplayed on the y-axis. Additionally, the number of particles isweighted by the volume of the average particle in the bin. The y-axis ofthe plot represents the volume fraction of the particles in the bin. Thetotal volume of particles in a unit volume of colloid is the sum of thevolume fraction in each of the bins. Ideally the histogram is plotted asa bar graph, which represents the width of the bins, but only the heightof the bars is plotted against the mean bin size as a two dimensionalplot.

A “concentration” of particles refers to the volume occupied byparticles in a unit volume of colloid.

A “wavelength” refers to the distance, measured in the direction ofpropagation of a wave, between two successive points in the wave thatare characterized by the same phase of oscillation.

The term “on line” refers to measurement of a process with automatedsampling and sample transfer to an automated analyzer.

The term “in line” refers to measurement of a process where the sampleinterface is located in the process stream.

The term “at line” refers to measurement of a process involving manualsampling with local transport to an analyzer located in themanufacturing area.

The term “small particles” as used herein, refers to particles within arange of about 5 nm to about 50 μm or to any subrange of this sizerange.

Embodiments of the present invention determine a particle sizedistribution for a sample of small particles by comparing a measurementvector and a reference matrix composed of reference vectors. Themeasurement vector represents the measured extinction spectrum of adilute colloid composed of the sample particles dispersed in a fluid.The reference vectors each represent the extinction spectrum of areference dilute colloid composed of particles of the same particlematerial as the particles of the sample and having a respective singleparticle size. The reference vectors are typically pre-computed, but mayalternatively be obtained empirically or experimentally.

The particles in the dilute sample colloid may be in the solid, liquid,or gaseous (vapor) phase, and the medium they are dispersed in may be inthe solid, liquid or gaseous (vapor) phase. The particles may be anymaterial for which the complex refractive index is known or can bemeasured over the ultraviolet (UV) to near infrared (NIR) wavelengthrange. The range of solid particles which can be measured includesceramics, metals, and polymers. The range of colloids from whichparticles can be measured includes dispersions, aerosols and emulsions.Exemplary fluids include liquids and gases, including, but not limitedto: water, air, alcohols, solvents, oils, or any other liquid or vapor.

When a broad particle size distribution is present, post-processing isperformed to remove noise generated by large particle sizes. In anycase, smoothing of results can be applied to smooth out spikiness thatmay be present in a particle size distribution due to artifacts of theinversion algorithm.

Mie Scattering

Embodiments of this invention include methods for efficiently andquickly computing the particle size distribution of a sample of smallparticles using linear equations, the measurement vector and thereference matrix. The extinction spectra represented by the measurementvector and reference matrix are essentially Mie scattering spectra.Bohren and Huffman, Absorption and Scattering of Light by SmallParticles, Wiley-VCH, 1983, pp. 318-319, which is hereby incorporatedherein, by reference thereto, describe the basic theory of Miescattering and how it can be used to measure the particle size ofparticles in a monodisperse sample.

FIG. 1 is a schematic diagram of a spectrophotometer 10 that may be usedto measure the extinction spectrum of a dilute colloid comprising asample of small particles by spectrophotometry. One non-limiting exampleof a spectrophotometer that may be employed for such purposes is theAgilent 8453 UV/Visible Spectrophotometer (Agilent Technologies, Inc.,Santa Clara, Calif.). In this example, spectrophotometer 10 includes alight source that incorporates both an ultraviolet light source 12, suchas a deuterium arc lamp, and a near-infrared-visible light source 14,such as a tungsten lamp. By using two different light sources generatinglight in different wavelength ranges, the smaller-sized particles in aparticle size distribution scatter more in the shorter wavelength light(UV spectrum) more while the larger-sized particles in the particle sizedistribution scatter the longer-wavelength light (near-IR—visible) more.This differential scattering is effectively wavelength divisionmultiplexing, so that the spectra due to the smaller-sized particles aredistinguished from the spectra due to the larger-sized particles by theshorter-wavelength extinction spectrum and the longer-wavelengthextinction spectrum that are generated. Optics 16 collimate the lightreceived from sources 12 and 14 into a beam 18 that is focused onto asample, which, in the example shown, is contained in a sample cell 20. Ashutter mechanism 22 may be provided which is actuatable to controlwhether the beam 18 is incident on the sample or is prevented therefrom.In one example, sample cell 20 is a cuvette having plane parallel faces21 and 23 formed of polished, fused quartz.

Additional optics 24 are provided on the output side of the sample cell20 to direct an output beam 26 of light that is transmitted through thesample to slit 28. Light that passes through slit 28 is reflected offgrating 30, which spreads beam 26 into its constituent wavelengthcomponents that are detected by a photodiode array 32. In the Agilent8453 UV/Visible Spectrophotometer, the photodiode array is composed of1024 photodiodes. Other array detectors may be alternatively used, suchas, but not limited to a charge-coupled device (CCD) or complementarymetal-oxide semiconductor (CMOS) detector. Alternatively, aspectrophotometer 10 may not employ the photodiode array 32, but rather,scan the output light 26 with a single detector by rotating the grating30 to enable the single detector to detect various wavelengths in thewavelength range of the output light 26. However, use of the photodiodearray 32 is preferred because it is faster.

A sample dilute colloid is formed from a sample of small particles(e.g., within a range of about 5 nm to about 50 μm) in a suitable fluid(e.g., water, air, oils, alcohols, solvents, etc.). The sample dilutecolloid is put in the sample cell 20 to be analyzed. Analysis of staticsamples of particles in a liquid or gas can be analyzed. For analysis ofsmall particles in a liquid, sample cell 20 may alternatively comprisesa flow-through cell. For analysis of small particles in a gas, a streamof the sample dilute colloid may be flowed into the sample cell 20.Alternatively, for analysis of small particles in a gas such as anaerosol, the aerosol can be flowed through the open space in the system12 between the light sources 10, 12 and the detector 32, e.g., in thespace occupied by the cell 20 in FIG. 1, as the cell 20 can be removedin this case. Alternatively to measuring particles in a flowing stream,measurements of a batch process can be performed, wherein samples aretaken from the batch process and measured at line. An extinctionspectrum of the (or each, in the case of batch processing) samplecolloid is measured with the spectrophotometer 10. Collimated light 18passes through the sample cell 20, where the sample colloid attenuatesthe light by scattering and absorption. The remaining light 26 isspectrally dispersed by diffraction grating 30 onto photodiode array 32,which converts light intensity to an electrical signal that isrepresentative of the extinction spectrum of the sample colloid. Theelectrical signal is processed to calculate the particle sizedistribution of the sample particles. The measured extinction spectrumis represented by a measurement vector that can then be used todetermine the particle size distribution of the sample particles, aswill be described below.

The Mie scattering theory (Gustav Mie, 1908) for a dilute colloidcomprising spherical particles provides a means to calculate thescattering component and the absorption component of the extinctionspectrum of a dilute colloid in which the volume concentration of theparticles in the colloid, the particle radius a, the length of the cellcontaining the colloid, the particle refractive index, and the fluidrefractive index are all known. The Mie scattering theory computes twoquantities, the absorption cross-section C_(abs) and the scatteringcross-section C_(sca). These cross sections add together to produce athird quantity C_(ext), the extinction cross-section. The extinctionspectrum referred to herein is the extinction cross-section at each of anumber of wavelengths in a range of wavelengths. All three crosssections have units of area and represent the effective cross-sectionalarea of the scattering, absorbing or scattering and absorbing sphere.Light that is lost to scattering is lost by radiation in directions thatare outside the range of angles of the light that the spectrophotometeris capable of measuring. Light that is absorbed is captured by theparticles and ultimately gets turned into heat. Particles that aretransparent at a particular wavelength, such as silica particles, have azero absorption cross-section at that wavelength. In the presentinvention, the extinction cross section C_(ext) is used, as it isirrelevant by what means the light is lost during the measurement.

The “relative cross section” is the extinction cross-section divided bythe physical area of the spherical particle, and is definedmathematically by:

$\begin{matrix}{Q_{ext} = \frac{C_{ext}}{\pi \; a^{2}}} & (1)\end{matrix}$

where α is the particle radius. Q_(ext) is a unitless quantity that is ameasure of the extinction efficiency. The present invention does notdistinguish between light that is lost by absorption and light that islost by scattering, but only measures extinction. If Q_(ext) is lessthan 1, then the apparent size of the particle is less than its physicalsize.

As long as the sample colloid comprising the sample particles is dilute(so that multiple scattering effects do not need to be considered in thecalculations) then the transmittance of a cell 20 (power out divided bythe power in) containing a sample colloid having a particle density of Nparticles per unit volume is:

T=e^(−αL)  (2)

where α=NC_(ext) and L is the cell length. The spectrophotometer 10measures attenuation, which is the negative logarithm of thetransmittance, expressed as:

AU=αL×Log₁₀ e=NC _(ext) Log₁₀ e×L  (3)

where AU stands for absorbance units, the numbers computed by thespectrophotometer. The particle density N can be expressed in terms of avolume fraction C_(v), wherein the number of particles per unit volumetimes the volume of a single particle is the volume fraction occupied bythe particles. This is expressed mathematically in the following way:

$\begin{matrix}{C_{v} = {\frac{4}{3}\pi \; a^{3}N}} & (4)\end{matrix}$

In other words, the number of particles N per unit volume times thevolume (4/3πa³) of a single particle is the fraction of the volume C_(v)occupied by the particles.

Combining equations (1), (3), and (4) gives an expression for theattenuation per unit length:

$\begin{matrix}{\frac{AU}{L} = {{\frac{3{Log}_{10}e}{4a}C_{v}Q_{ext}} = {\frac{3{Log}_{10}e}{4a}{C_{v}\left( {Q_{sca} + Q_{a\; {bs}}} \right)}}}} & (5)\end{matrix}$

where Q_(sca) is a unitless quantity that is a measure of the scatteringefficiency and Q_(abs) is a unitless quantity that is a measure of theabsorption efficiency.

Equation (5) can be used to compute optical attenuation from theextinction cross-section obtained from the Mie calculation. The presentinvention provides solutions for computing the particle sizedistribution from the measured extinction spectrum. That is, theparticle size distribution can be computed using the measured extinctionspectrum as an input. This problem belongs to a class of problems knownas “inverse problems”, which are notoriously difficult to solve. Thepresent invention uses any of several linear programming methods tosolve this problem.

Before the inverse problem is described, equation (6) firstmathematically describes how the extinction spectrum of particles in adilute colloid can be calculated using a matrix M of reference vectorsand a concentration vector. A matrix equation of the form of equation(6) below can be used to calculate the extinction spectrum of a dilutecolloid composed of a given fluid and given particles having a givenparticle size distribution. The particle size distribution is composedof a discrete set of particle sizes a, with a different particleconcentration (volume concentration) for each particle size:

M·c=s  (6)

where M is a matrix containing the reference vectors (see the right handside of equation (8) below), one for each particle size or small rangeof particle sizes, c is a column vector composed of a respectiveparticle concentration for each particle size in the range of particlesizes, and s is the computed extinction spectrum, also a column vector.That is, the calculated extinction spectrum is composed of a calculatedextinction for each wavelength in the range of wavelengths over whichthe calculation is performed. Each calculated extinction covers adifferent wavelength or range of wavelengths of the extinction spectrum.

The particle size distribution and particle concentrations can bedetermined from a measured extinction spectrum by linear methods, suchas when the PSD and thus the concentration vector c are not known, andthe extinction spectrum is measured and is represented by a measurementvector s. It is noted that the extinction cross-section is a function ofparticle radius, and particle and fluid refractive indices (refractiveindices of the sample particles and fluid constituting the dilutecolloid). Therefore equation (5) can be expressed in the following form:

$\begin{matrix}{\frac{AU}{L} = {C_{v}{f\left( {a,{n_{p}(\lambda)},{n_{f}(\lambda)}} \right)}}} & (7)\end{matrix}$

Where n_(p) and n_(f) are the wavelength-dependent refractive indices ofthe particles and the fluid, respectively. For a discrete set ofmeasurement wavelengths └λ₁, λ₂, λ₃, . . . λ_(n)┘ such as that of thespectrophotometer 10, equation (6) can be written as the product of avector representing an extinction spectrum and a particle concentration,which is a scalar quantity.

[AU(λ₁),AU(λ₂),AU(λ₃) . . . AU(λ_(n))]=C_(v)L[ƒ(α,λ₁),ƒ(α,λ₂),ƒ(α,λ₃), .. . ƒ(α,λ_(n))]  (8)

where the function ƒ now implicitly includes the materials propertiesfor a particular dilute colloid, i.e., the materials properties of thesample particles and the fluid that constitute the colloid.

Reference Matrix

The matrix M described above is referred to herein as a referencematrix, and includes columns of reference vectors. Each reference vectorrepresents a reference extinction spectrum for a respective referencecolloid. The respective reference colloid is composed of referenceparticles of the same particle material as the sample particles andhaving a respective single discrete particle size or small range ofsmall particle sizes. The reference particles are dispersed in the samefluid as that which constitutes the fluid of the sample dilute colloid.In one embodiment, the reference extinction spectrum represented by eachreference vector is composed of a wavelength value and an extinctionvalue measured at that wavelength value by spectrophotometer 10 for areference dilute colloid having the respective single particle size. Inanother embodiment, the reference extinction spectrum represented byeach reference vector is composed of a wavelength value and anextinction value computed at that wavelength value for a referencedilute colloid having the respective single particle size. Inembodiments in which the wavelength values are conveyed independently ofthe reference vectors, the wavelength values may be omitted from thereference vectors.

Using equation (8), the particle size distribution measurement problemcan be expressed as:

$\begin{matrix}{{\begin{bmatrix}m_{11} & m_{12} & m_{13} & \cdots & m_{1m} \\m_{21} & m_{22} & m_{23} & \cdots & m_{2m} \\m_{31} & m_{32} & m_{33} & \cdots & m_{3m} \\\vdots & \vdots & \vdots & ⋰ & \vdots \\m_{n\; 1} & m_{n\; 2} & m_{n\; 3} & \cdots & m_{n\; m}\end{bmatrix}\begin{bmatrix}c_{1} \\c_{2} \\c_{3} \\\vdots \\c_{m}\end{bmatrix}} = \begin{bmatrix}s_{1} \\s_{2} \\s_{3} \\\vdots \\s_{n}\end{bmatrix}} & (9)\end{matrix}$

where the measurement vector representing the measured extinctionspectrum of the sample dilute colloid is substituted for the computedextinction spectrum s in equation (6). The concentration vector c issolved for to give the particle size distribution of the sampleparticles. The sample size distribution is composed of a particleconcentration for each of the various particle sizes.

Various different approaches/algorithms can be used to solve for theconcentration vector c, some having more drawbacks than others. Oneapproach computes the inverse of M and then solves for c by multiplyingthe inverse of M by s. This approach works on the condition that thematrix M is a square non-singular matrix. In practice, this condition istypically not satisfied.

Other approaches attempt to find a concentration vector c that minimizesthe least-square error in the fit error between the measurement vectorand the extinction spectrum calculated using the reference matrix andconcentration vector, as described in the following. As one example thepseudo-inverse of the matrix M is computed, and then c is solved for bymultiplying the pseudo-inverse of M by s. This method is based onSingular Value Decomposition of matrix M. The advantages of thisapproach are that it does not require a square non-singular matrix.Although this works well in theory, in practice measurement errors,noise, and uncertainties in refractive indices renders this approachless than desirable, as it sometimes computes negative values for theconcentrations in some size bins/particle sizes.

Another method uses the “NMinimize” routine in the MATHEMATICA® computerapplication (Wolfram Research). NMinimize is a numerical optimizationroutine that finds the concentration vector which minimizes the leastsquare fit error in the computed spectrum using the Nelder Meadealgorithm. Non-negative constraints are applied to the concentrationvector elements. This allows equation (6) to be solved with constraintson the concentration vector c. The only constraint necessary is therequirement that the elements of the concentration vector c benon-negative.

Quadratic Programming is another approach that may be used. Thequadratic programming technique is used in a similar fashion to theNMinimize algorithm, with the exception that it requires matrix M to bepositive definite. While this condition is not true for matrix M ingeneral, matrix M may be forced to be positive definite by altering thematrix M slightly. This method works, but altering the matrix M alsoalters the calculation results, and therefore this is not an idealapproach.

QR decomposition is the preferred approach to use with the presentinvention. In linear algebra, the QR decomposition (also called the QRfactorization) of a matrix is a decomposition of the matrix into anorthogonal and a triangular matrix. The QR decomposition is often usedto solve the linear least squares problem. The QR decomposition is alsothe basis for a particular eigenvalue algorithm, the QR algorithm. TheQR decomposition is an intermediate step in solving the non-negativeleast squares problem solved by the methods described herein. QRdecomposition can be implemented with positive concentration constraintson the concentration vector c. The measured spectrum vector and thereference matrix are inputted and the QR decomposition algorithm findsthe optimum concentration vector with the constraints thatconcentrations are non-negative. The QR decomposition, NMinimize, andQuadratic programming are all used in similar ways, although the detailsof the methods are different. NMinimize is too slow and it isproprietary code. Quadratic programming distorts the result, as notedabove. QR decomposition is the preferred option among these three. Theonly potential issue found when using QR decomposition is that it insome instances may tend to produce a narrower particle size distributionthan is physically reasonable.

Other constrained least-squares solutions that may be used include, butare not limited to those identified in Golub, Matrix Computations,3^(rd) edition (which is hereby incorporated herein, in its entirety, byreference thereto): Ridge Regression, Equality Constrained leastsquares, the method of weighting or LS minimization over a sphere.

FIG. 2 is a flow diagram 200 showing an example of a method forcomputing the particle size distribution of a sample of small particlesaccording to an embodiment of the present invention. At 202 a referencematrix M of pre-computed or pre-measured reference vectors is provided,each reference vector representing a discrete particle size of aparticle size distribution of particles contained in a referencecolloid, each reference vector representing a reference extinctionspectrum over a predetermined wavelength range.

Each reference vector (column) of the reference matrix M includes areference extinction spectrum calculated or measured for a respectivereference colloid comprising particles of a single, known particle size.The reference colloid is a dilute colloid of specified concentration.The reference particles are particles of a particle material havingknown values of refractive index at various wavelengths within thewavelength range over which the measure extinction spectrum is measured.Typically, but not necessarily, in each reference vector, each elementis composed of a wavelength and an extinction value. The extinctionvalue is calculated at the wavelength for the respective referencecolloid or is measured at the wavelength by spectrophotometricmeasurement of the respective reference colloid.

At 204, a measurement vector representing measured extinction values ofa sample colloid comprising the sample particles is provided. As notedabove, the sample colloid is a dilute colloid. The particles in thesample colloid are the same material as some or all of the particles ofthe reference colloid(s) used to generate the reference matrix. Themeasured extinction values are those measured by spectrophotometricmeasurement at discrete wavelengths. The measurement of the sampledilute colloid may be performed online, in real time, with thespectrophotometer inputting the measurement vector into the system forcalculation of particle size distribution. Because of the speed of thesemethods, particle size distribution and particle concentrationcalculations can be performed for sample particles that constitute partof a flowing sample colloid. The sample colloid may be obtained, forexample, from a process stream. The spectral measurement used to providethe measurement vector is fast, on the order of a second, when using aspectrophotometer such as the Agilent 8453. The computation of theparticle size distribution and particle concentrations from themeasurement vector takes only a few additional seconds when using thefast matrix methods for linear least squares deconvolution describedherein. Advantageously, this allows real time calculations of particlesize distributions and particle concentrations. Such real timecalculations can be performed for sample particles that constitute partof a static sample colloid, as well as for sample particles thatconstitute part of a flowing sample colloid. For example, the presentmethods can be performed iteratively to provide particle sizedistributions and particle concentrations that can be analyzed tomeasure crystal growth, or a reduction in the particle sizes ofparticles in a process stream, or in a still sample. These iterativecalculations can be performed after every passing of a predeterminedtime interval, e.g., every five seconds, every two and a half seconds,or even every one second. Of course, other predetermined time intervalsnot mentioned here, and greater than one second can be set. If themeasurement time of the spectrophotometer can be reduced, this wouldallow the present methods to be performed iteratively over a timeinterval of less than one second. The results of the calculations frommultiple iterations can be compared with one another and/or against apredetermined threshold value that is meaningful to a process. Forexample, a chemical process may be designed to produce particles havinga predetermined size. Accordingly, a threshold can be set for apredetermined concentration of particles at that predetermined size.When, upon comparing calculated particle size distribution and particleconcentration results, or comparing the concentration of a particularparticle size of interest only with a predetermined concentrationthreshold for that size (which can simplify and thus even furtherincrease the calculation processing) the comparison identifies aparticle concentration of the particle size of interest (predeterminedsize) that is equal to or greater than the predetermined concentration,than the process can be configured to generate an alert to indicate thatthe process has achieved a goal. Similar monitoring can be performed toidentify when a process has reduced the sizes of particles to apredetermined particle size or to achieve a predetermined particle sizedistribution.

In one example, the extinction spectrum of a process stream is measuredusing, for example, a spectrophotometer 10. First an initial measurementis made of the extinction spectrum of the fluid component (that is,without the particles to be measured) of the colloid in the uv-NIRwavelength range. This measurement of a “blank” spectrum using a samplecell containing the fluid without the particles provides a referenceattenuation of the medium component alone. It also calibrates thespectrophotometer to compensate for changes in the lamp spectra and thevarious losses and gain in the entire system. Next, the extinctionspectrum of light transmitted through the sample dilute colloidcontaining the sample particles is measured in the same way that theinitial measurement was made. The second measurement with particlesgives an accurate measure of the excess attenuation due to theparticles, according to the following. The resulting spectrum obtainedby dividing the measured extinction spectrum of the sample colloid bythe initial extinction spectrum represents wavelength-dependence of thetotal light-attenuating effect of all the sample particles in the pathof collimated beam 18 (FIG. 1) through the sample colloid. The measuredspectrum is represented by a measurement vector. This measurement vectoris then deconvolved, such as by QR decomposition, using a least squaresfit, such as a non-negative least squares (NNLS) fit, to minimize thedifference between the measurement vector and the calculated extinctionspectrum obtained from the concentration vector and the reference matrixM in a manner as described above. A particle size distribution vector isgiven by the concentration vector providing the best least-squares fit.The elements of the concentration vector can be used to give theparticle concentration of each size of particle present in the sample.

That is, the particle size distribution and particle concentrations ofthe sample particles in the sample colloid are determined at 206, usinglinear equations, the reference matrix M and the measurement vector in amanner as described above. Of course, the present invention is notlimited to providing the particle size distribution, as the presentmethods can be used to determine only the particle size, determine onlythe concentration of a particle size of interest, or to determine anysubset of the information provided by the full particle sizedistribution with particle concentrations. Optionally, the particle sizedistribution and concentrations of particles in the sample colloid canbe output at 208. Additionally or alternatively, further processing canbe performed, including, but not limited to comparing one or moreparticle size distributions and/or one or more of the concentrations toa threshold value and/or to each other.

For example, if the sample particles in the sample colloid are known tobe only particles of particle material “A”, the reference extinctionspectra for reference colloids respectively comprising particles ofparticle material “A” are computed or measured over a range of particlesizes of interest (e.g., a range of particle sizes within which theparticle sizes of the sample particles are expected to lie) and thereference extinction spectra are included in the reference matrix M.Each reference colloid is a dilute colloid comprising particles having aknown single particle size. The concentration vector given by theleast-squares fit of the measurement vector to the extinction spectrumcalculated using the reference matrix and concentration vector gives theparticle concentration of each size of sample particle present in thesample.

Extinction spectra can vary drastically depending on the particle sizesof the particles measured. FIG. 3 shows at 310 and 320 the calculatedextinction spectra of two exemplary sample colloids comprisingpolystyrene particles having particle sizes of about 100 nm and 5 μm,respectively, at a particle concentration of about 0.1 volume percent.The extinction spectra are calculated over a wavelength range of about190 nm to about 1100 nanometers. The extinction spectra 310 and 320 havebeen overlaid in FIG. 5 for comparison purposes, but each was calculatedseparately.

Calculating the reference extinction spectra of monodisperse referencecolloids each comprising particles of a single particle size isstraightforward and each calculation produces a unique result. However,the inverse operation, as noted above, is not straightforward, i.e., thecalculation of particle size distribution from a measured extinctionspectrum does not necessarily result in a unique solution. For example,colloids having different combinations of different size particles cancreate the same measured extinction spectrum when measured, as spectralfeatures tend to “wash out” when particles having large numbers ofdifferent particle sizes are present. Also, it is difficult to measurethe particle size distribution of particles that are much smaller thanthe wavelengths used to measure the extinction spectrum, e.g., particleshaving a size approaching the Rayleigh scattering limit. Theabove-described method of computing particle size distribution iswell-adapted to measuring particle sizes in the range of about 10 nm toabout 15 μm. The above-described method provides the most probablesolution to the calculation of the particle size distribution andparticle concentration, by finding the particle size distribution whoseextinction spectrum is the best least squares fit to the measuredextinction spectrum.

When determining a reference matrix M, the range of particle sizes overwhich particles are expected to be present in the sample is divided intoa number of discrete particle sizes. Each particle size covers arespective portion of the entire range of particle sizes over which theparticle size distribution is to be measured. Typically this divisionwill result in hundreds of particle sizes. Typically, the entire rangeof particle sizes is divided into particle sizes on a logarithmic scale.

For each particle size, the respective reference extinction spectrum isgenerated by computing or measuring a reference extinction value at eachone of multiple discrete wavelengths over the range of wavelengths inwhich the measured extinction spectrum is measured. In an example,respective extinction values are measured or computed at wavelengthsdiffering by 1 nm steps or differing by increments each of which is aninteger multiple number of nanometers. In at least one embodiment,reference extinction spectra are computed or measured at wavelengthsdiffering by increments of 4 nm, which gave an acceptable compromisebetween resolution and operational speed. Thus, there are also typicallyhundreds of rows in the reference matrix M, each corresponding to onewavelength. In equation (9) above, referring to the reference matrix M,n is the index number of the particle size, m is the index number of thewavelength at which the extinction value is measured, and each column ofthe matrix M represents a reference vector representing the referenceextinction spectrum of a reference colloid comprising particles of arespective particular size.

After a measurement vector is provided, as described above with regardto FIG. 2, 204, the least squares error is calculated between thereference extinction spectrum (obtained by multiplying a value ofconcentration vector c by matrix M) and the measured extinction spectrum(represented by the measurement vector) and the least squares error isminimized by varying the calculated spectrum (e.g., by varying thevalues of the concentration vector c, and thereby finding the best fitsolution where a global minimum in the least squares error results. Anextinction spectrum is computed by multiplying the matrix M by theconcentration vector. Then the process finds the concentration vectorthat minimizes the error between the measured extinction spectrum(matrix M) and the computed extinction spectrum. The individualconcentration elements of the concentration vector can be varied bytrial and error to find the minimum least squares error, but thenon-negative least squares algorithm based on a modified QRdecomposition ensures that the solution does not get trapped in a localminimum. It is a deterministic approach that determines the globalminimum solution. The matrix is computed assuming an arbitrary particleconcentration (volume concentration), such as 1%. The actualconcentration is then 1% times the concentration found by thenon-negative least squares global minimum solution.

FIG. 4 illustrates a plot 400 of the least squares error values againstthe concentration vector c. As can be seen there are several localminima 402, 404, 406 in the plot. The present invention locates theglobal minimum 404 as the most probable solution for the measurement ofthe particle size distribution and concentrations of particles in themeasured extinction spectrum represented by the measurement vectorhaving been used with the reference matrix M to compute variousconcentration vectors c. Thus, in the example shown in FIG. 4, the mostprobable solution to the particle size distribution calculation is thatgiven by the value of the concentration vector c plotted at 408. Thecalculated extinction spectrum obtained by multiplying this value of theconcentration by the reference matrix M most closely matches themeasured extinction spectrum.

The other local minima 402 and 406 might be identified as solutions insome prior art systems, due to the nature of different combinations ofparticle sizes sometimes producing measured extinction spectra that arevery similar, as noted above. For example, a sample dilute colloidhaving 100 nm particles and 300 nm particles, when the measuredextinction spectrum represented by the measurement vector is comparedwith the calculated spectrum obtained by multiplying the referencematrix M with various concentration vectors c, may provide alternativesolutions, such as a solution at local minimum 402 indicating that theparticles measured are all 200 nm particles, while the global minimum404 indicates that the particles measured were 100 nm and 300 nmparticles. By consistently finding the global minimum 404, the presentsystem ensures that the most probable particle size distribution resultis attained.

Artifacts

The results of a particle size distribution calculated by the methodsdescribed above are typically quite spiky, especially when a significantamount of noise is present in the system. For example, FIG. 5A shows aparticle size distribution 510 calculated from a measurement vector andreference matrix according to the techniques described above when about1% noise was present in the system. The term “noise” here refers to theto the root mean square deviation of the noise waveform which is addedto the signal, calculated as a percentage of the signal, as is commonlyused in computing the signal-to-noise ratio, which is the inverse of the1% noise referred to here. That is 1% noise means a 100 to 1signal-to-noise ratio. In many usages, a 1% noise is not considered verysignificant and it can be difficult, as a practical matter to achievelower noise levels. Because the 1% noise level does have a significanteffect on the computed particle sizes, it is advantageous to postprocess these computed particle sizes.

FIG. 5B shows the particle size distribution 520 calculated from thesame measurement vector and reference matrix used to calculate theparticle size distribution 510, but where much less noise (substantiallynone) was present in the system. Because in practice, there is typicallyalways some noise present, one or more smoothing techniques are appliedto the calculated particle size distribution to smooth out the spikinessof the result. Spiky results are difficult for a user to interpret,since spiky results do not allow the user to simply locate a value alongan axis of the plot by simply cross referencing it with a value alongthe other axis. At such location, there may be a discontinuity in theplot, or the value at such location may be substantially higher or lowerthan what it would be, if the spikiness were removed. Similarconsiderations apply to the interpretation of the particle sizedistribution or particle concentrations by a subsequent system. Furtherdiscussion of smoothing techniques that can be applied to reduce thespikiness of the particle size distribution result is provided belowwith reference to FIG. 10A.

Another noise artifact can occur when the particle size distributions ofparticles is broad. A broad particle size distribution is defined as aparticle size distribution in which the number of non-zero particlesizes for which the particle concentration is greater than zero isgreater than a predetermined percentage of the total number of particlesizes in the reference matrix M and therefore are present in theparticle size distribution.

Inversion of the reference matrix resulting from a broad particle sizedistribution (PSD) spectra using the non-negative least squares (NNLS)algorithm can induce a noise artifact that causes the resulting particlesize distribution to indicate an excess of large particles. That is, thecomputed particle size distribution will indicate the presence of largeparticles that are not really present in the sample. This artifactpossibly arises because particles larger than the measurement wavelengthdo not contribute much detail to the extinction spectrum. A broaddistribution of large particles simply looks like a nearly constantoffset in absorption units over the wavelength range. When the particlesare much larger than a wavelength, then they simply cast shadows on thedetector. There is no spectral dependence. The spectrum is flat.Therefore no distinction can be made between one really large particleand two particles that each have half the cross-sectional area of thereally large particle. Any spectrum that is flat can be fit by adding adistribution of large particles. The problem is that there is more thanone solution to this “ill-posed” problem. Accordingly, the presenttechniques find the solutions with the minimum number of filled bins,which eliminates the broad distribution of large particles. Accordingly,when a broad distribution of particles is present, the present inventionprovides a method (a large particle artifact removal application) toeliminate this large particle noise. The large particle artifact removalapplication is described in detail below after a description of theexample shown in FIG. 6.

A sample colloid including silicon nitride particles was prepared andits particle size distribution was measured according to theabove-described methods. The silicon nitride was a NIST (NationalInstitute of Standards) reference material 659, which is used forcalibrating a sedigraph, an instrument that measures particle size bysedimentation. According to NIST, the particles should have thefollowing cumulative weight distribution, as shown in Table 1 below.

TABLE 1 Cumulative Weight Percentile Certified Value (μm) Uncertainty(μm) 10 0.48 0.10 25 0.81 0.10 50 1.43 0.10 75 2.08 0.11 90 2.80 0.13

The silicon nitride sample was prepared and then diluted toapproximately 0.02165% by weight or 0.00679% by volume in water toprovide the sample dilute colloid. The dilution was performed to enablesufficient light to be transmitted through the 1 cm path length cell 20to make an extinction spectrum measurement with the 8453spectrophotometer. A reference matrix M was computed using known datafor the particle sizes, the refractive index of silicon nitride, and therefractive index of water. The resulting matrix M was a 911 row by 911column matrix. The 911 columns covered a particle size range from 10 nmto 15 μm. The measurement vector representing the measured extinctionspectrum of the sample colloid was input along with reference matrix Mto a process that solves for the concentration vector c via thenon-negative least squares method in a manner as described above. ThePSD 600 that produced the best matching spectrum is shown in FIG. 6.

While the NIST data in Table 1 indicate that the sample has fewparticles greater than 3 μm in diameter, the raw particle sizedistribution 600 indicates that the bulk of the particles 602constituting the sample are larger than 3 μm. Additionally, it can beobserved that the total concentration of particles in the result 600 istoo high at 0.0322% by volume rather than the prepared 0.00679%.However, the fit to the measured spectrum is quite good with an RMS fiterror of 0.11%.

The large particle artifact removal application provides a method ofconstraining the particle size range used in the inversion. In theexample shown in FIG. 6, reducing the upper particle size range boundfrom 15 μm to about 3 μm had almost no effect on the fit error eventhough the particle size distribution changed significantly. Theseresults are shown in FIG. 7. The RMS fit error 700 only slowly increasesfrom about 0.11% when the particle size upper bound is set to 15 μm 702,to about 0.15% when the particle size upper bound is set to 1.5 μm 704.As the particle size upper bound is reduced below 1 μm 706, the fiterror increases rapidly.

After constraining the maximum particle size to 1.5 μm, the resultingparticle size distribution was smoothed, with a 3% factor to reduce thespiky particle size distribution result, the particle size distribution800 shown in FIG. 8 was produced, and the total particle concentrationresult was calculated to be 0.0040% by volume. “Smoothing with a 3%factor” refers to the number of bins in the particle size distribution(the number of elements in the concentration vector). Smoothing isperformed by convolving the particle size distribution with a Gaussiankernel with width equal to 3% of the total number of bins. Thus, ifthere are 100 bins, then the convolution kernel has a width of 3. Thissmoothing function is a Gaussian blurring function, similar to theGaussian blur function provided in Adobe PHOTOSHOP, except it is a onedimensional blurring function, rather than the two-dimensional blurringfunction provided in PHOTOSHOP. An even better match to the known totalparticle concentration was calculated when the particle size upper boundwas increased to 3 μm.

As shown, when the above-described calculation methods result in broadparticle size distributions, the raw particle size distribution resultsshould be post-processed to eliminate large particle noise. The largeparticle artifact removal application provides one solution to reducingor removing large particle noise. Other post-processing techniques maybe alternatively applied. For example, the particle size distributionresults can be fit to a log-normal distribution to reduce or eliminatelarge particle noise.

Another artifact to address is one that can occur in the extinctionspectra measured by the spectrophotometer, and is referred to as forwardscattering artifact. The above-described scattering attenuation modelassumes that all light that is scattered is lost. For the most part thisis true. However a small portion of the light that is scattered in theforward direction is captured by the receiving optics 24. The amount offorward scattering depends on the ratio of the particle size to theilluminating wavelength as well as the capture angle of the receivingoptics.

When the extinction spectrum for large particles is measured, theforward scattering artifact reduces the total attenuation observed,particularly at short wavelengths, where the size-to-wavelength ratiobecomes large. An example of this reduction is shown in the extinctionspectrum plot 900 in FIG. 9A, where the attenuation in the portion 902of the plot 900 of the measured extinction spectrum of 5 μm polystyreneparticles has been reduced due to forward scattering. FIG. 9Billustrates a plot 910 corrected for forward scattering artifact byapplying a forward scattering correction factor to plot 900. Plot 900 isalso shown for comparison.

A method for calculating attenuation in absorbance units per unit length(AU/L) as a function of the relative extinction cross-section Q_(ext)was described above. A method for calculating the forward scatteringcorrection factor, represented by ΔAU/L is now described. This forwardscattering correction factor is subtracted from the raw absorbance perunit length to give the forward-scattering-free AU/L to get a moreaccurate scattering model. Assuming that the spectrophotometer admits acone of scattered light with half angle β, then it can be shown that thecorrection factor in AU per unit length L is

$\begin{matrix}{{\frac{\Delta \; {AU}}{L} = {\frac{3\log \; e}{16\pi^{3}}\frac{C_{v}}{a^{3}}\frac{\lambda^{2}}{n_{f}^{2}}2\pi {\int_{0}^{\beta}{{{X(\theta)}}^{2}{\sin (\theta)}{\theta}}}}},} & (7)\end{matrix}$

where |X(θ)|² is the magnitude squared of the vector scatteringamplitude as defined in Bohren and Huffman.For small particles and small limiting angles β, then equation 7 reducesto the approximate expression:

$\begin{matrix}{{\frac{\Delta \; {AU}}{L} \cong {\frac{3\log \; e}{16\pi^{3}}\frac{C_{v}}{a^{3}}\frac{{{S_{1}\left( 0^{o} \right)}}^{2}\lambda^{2}}{n_{f}^{2}}{\Delta\Omega}}},} & (8)\end{matrix}$

where it is assumed that |X(θ)|²≅|X(0°)|²=|S₁(0°) |² for all θ≦β theinstrument acceptance angle. In addition the solid angle ΔΩ of theconical beam of half angle β is given by

ΔΩ=2π(1−cos(β))  (9)

If β is 0.8 degrees, then ΔΩ=0.000612 sterradians. The solid angle ΔΩ isa fixed property of the spectrophotometer.

Post Processing

FIG. 10A is a flow diagram illustrating post processing methods that canbe performed after computing a raw particle size distribution of asample of small particles. These methods can be implemented forautomatic processing by the system, such as by one or more computerprocessors incorporated into the system, to produce the most probableparticle size distribution and particle concentration results for asample that constitutes part of a sample dilute colloid whose extinctionspectrum has been measured by spectrophotometry. At 1002, the rawparticle size distribution (PSD) is calculated according to methodsdescribed above. For example the NNLS algorithm may be applied to themeasurement vector and a reference matrix M to calculate the particlesize distribution and particle concentrations of the particles. At 1004it is determined whether the raw PSD is a broad particle sizedistribution. A broad particle size distribution is defined as a rawparticle size distribution in which the number of particle sizes forwhich the particle concentration is greater than a threshold particleconcentration, in greater than a predetermined percentage of the totalnumber of particle sizes in the reference matrix M. In one example, abroad particle size distribution is defined as a raw PSD having aparticle concentration greater than a threshold particle concentrationof zero in greater than 3% of the total number of particle sizes.

When it is determined at 1004 that the PSD is a broad PSD, then postprocessing to eliminate large particle noise is carried out. Forexample, the large particle artifact removal application is applied at1005 to remove reference vectors from the reference matrix thatrepresent spectra of reference particles towards the larger particlesize end of the particle range, until the fit error obtained using areduced-size reference matrix exceeds that using the original referencematrix M by a predetermined factor.

The process of removing reference vectors from the reference matrix canbe performed according to various different schemes. FIG. 10Billustrates a subroutine 1006-1009 as one example of carrying out theprocess described in 1005 in FIG. 10A, to iteratively remove the largestparticle size vector. At 1006, the column representing the largest sizeparticle is removed from reference matrix M to generate a new, smallerreference matrix. At 1007, the raw PSD is recalculated, using the new,smaller reference matrix resulting from eliminating from referencematrix M the column vector representing the largest particle size. TheRMS error in the fit between the measured spectrum and the spectrumcalculated using the reduced-size reference matrix and the recalculatedraw PSD of the current iteration is calculated at 1008. The current RMSfit error is compared with the original RMS fit error calculated in amanner described above, between the measured extinction spectrum and thecalculated extinction spectrum calculated using the original, full-sizereference matrix M and the raw PSD calculated at 1002. If the errorincrease percentage calculated by the difference between the current RMSFit Error and the original RMS fit error calculated using the full-sizereference matrix, divided by the original RMS fit error, is less than apredetermined percentage, then processing returns to 1006 to performanother iteration of 1006, 1007, 1008 and 1009, beginning with removingthe largest remaining particle size vector at 1006 to again reduce thereference matrix M. Once the RMS error difference between the currentRMS error and the original RMS error is greater than or equal to apredetermined percentage (in one non-limiting example, twenty percent),then the iterative process is ended (“No” at 1009) and the upperparticle size bound has been determined, as well as the smallestpractical reference matrix M to use in calculating the PSD.

As noted, this process is not limited to removing one bin at a time, asmore than one bin at a time may be removed to make the process faster.FIG. 1C illustrates an alternative, preferred process for large particleartifact removal, wherein 1056, 1007-1009, 1058, 1007-1008, 1060, 1062,1064 and 1018 are substituted for 1005 and 1018 of FIG. 10A in order topost process a broad PSD. Thus, when it is determined at 1004 that abroad PSD exists, a predetermined number or percentage of the totalnumber of bins are removed at 1056, where the bins removed are thelargest particle size bins. At 1007, the raw PSD is recalculated, usingthe new reduced-size reference matrix resulting from eliminating thecolumns representative of the largest particle size bins removed at1056. The RMS error in the fit between the measured spectrum and thespectrum calculated using the reduced-size reference matrix and thecalculated raw PSD of the current iteration is calculated at 1008. Thecurrent RMS fit error is compared with the RMS fit error of the RMS fiterror having been calculated, in a manner described above, using themeasured spectrum, the spectrum calculated using the original, full-sizereference matrix and the raw PSD calculated at 1002. If the errorincrease percentage calculated by the difference between the current RMSfit error and the original RMS fit error calculated using the full-sizereference matrix, divided by the original RMS fit error, is less than apredetermined percentage, then processing returns to 1056 to performanother iteration of 1056, 1007, 1008 and 1009, beginning with againremoving the predetermined number or percentage of the total number ofbins remaining constituting the largest remaining particle size bins at1056 to again reduce the reference matrix M.

If the error increase percentage at 1009 is not less than thepredetermined percentage, then at 1058 a predetermined percentage(rounded to nearest integer) of the number of bins removed in theprevious removal procedure are added back in to increase the size of thematrix M. The bins added back in are the smallest ones from the totalnumber of bins in the previous removal. The raw PSD is againrecalculated (1007) and the RMS Fit Error is again calculated (1008)using the latest revise matrix M. If at 1060, the error increase is lessthan a predetermined percentage, when comparing the current RMS fiterror to the RMS fit error and the original RMS fit error in a mannerdescribed above, then a predetermined percentage (rounded to the nearestinteger) of the number of bins that were added back in at 1058 areremoved at 1062. The bins that are removed are the largest ones from thegroup of bins previously added back in. The 1007, 1008 and 1060 areagain executed, and this loop is iterated until a “No” answer is reachedat 1060. When the error increase is not less than the predeterminedpercentage at 1060, then it is determined whether the iteration of 1060just prior to the current one found an error increase less than thepredetermined percentage. If Yes, then another iteration of 1058, 1007,1008 and 1060 are performed. If No, than processing proceeds to 1018 asindicated in FIG. 10B as well as FIG. 10A.

In one embodiment, the predetermined percentage is 50%. Thus, forexample, if there are originally 400 bins, then 200 bins are removed at1056, 100 bins are added back in at 1058, and 50 bins are removed at1062, etc.

When the PSD is determined not to be a broad PSD at 1004, then postprocessing for large particle size noise is not needed, and the particlesize distribution data is smoothed using at least one of four methods1018, 1020, 1022 and 1023. Method 1018, referred to as “curve fittingoptimization” fits multiple Gaussian curves to the PSD to smooth thePSD. The “Gaussian blurring” method at 1020 is another technique thatcan be used to smooth the PSD. The best single match technique forcesthe solution to find a single size bin of the reference matrix M thatgives the best fit, as an alternative to computing the PSD bynon-negative least squares calculations or any other least squarealgorithm or other algorithm described above for inverting the referencematrix M. The best single match approach forces all the bins but one tozero. Since a single-bin solution is not very likely, this solution issmoothed by one of two methods. The first smoothing technique fits aGaussian with three fitting parameters: the mean, the standarddeviation, and the concentration. The best RMS fit is achieved bystarting with the best single match solution as the mean, but allowingboth the mean and the standard deviation to wander. The simple smoothingalgorithm effectively broadens the single bin by a fixed standarddeviation (the 3% factor mentioned above determines the width).Alternatively a log-normal distribution can be used instead of aGaussian. Method 1022 uses the best single match technique and thensmoothes by the “curve fitting optimization” method 1018. Method 1023uses the best single match technique and then smoothes by Gaussianblurring 1020.

At 1024, when more than one of the above-described smoothing processes1018, 1020, 122 and 1023 is performed, the best fitting one of theresults provided by the smoothing processes performed is identified. Ifonly one of the smoothing processes is performed, the solution producedby that process is used. Assuming more than one of the smoothingprocesses is performed, the best fit is determined at 1024 by comparingthe RMS fit error for each of the methods and choosing the smallest. Theresult of the best fitting method is then selected and applied at 1026,1028, 1030 or 1032, depending upon which method is determined to providethe best fit, thereby providing the smooth particle size distributionand particle concentrations results. Optionally, the best fit solutioncan then be displayed, such as on a user interface, or otherwise outputfor use by a human user. Further optionally, in addition or alternativeto displaying, the best fit solution can be passed to subsequentprocessing for use, e.g., in controlling a process that generates theparticles subject to analysis.

After performing large particle noise suppression as described above,curve fitting optimization may be performed at 1018, and/or Gaussianblurring may be performed at 1020 on the PSD calculated using thereduced-size reference matrix resulting in the last iteration of 1006,1007, 1008 and 1009. At 1012, the better fitting result from theprocesses 1008 and 1010 is identified, when more than one of theseprocesses is performed. If only one is performed that it is the solutionthat is used. Assuming more than one is performed, the better fit isdetermined by comparing the RMS fit error for each of the methods andchoosing the method resulting in the smaller RMS fit error. The resultof the better fitting method is then selected and applied at 1014 or1016, depending upon which method is determined to provide the betterfit, thereby providing the smooth particle size distribution andparticle concentrations results. Optionally, the better fit solution canthen be displayed, such as on a user interface, or otherwise output foruse by a human user. Further optionally, in addition or alternative todisplaying, the better fit solution can be passed to subsequentprocessing for use, e.g., in controlling a process that generates theparticles subject to analysis.

Computing Particle Refractive Indices

Refractive index data for the wavelength range over which the spectraused by the particle size distribution measurement processes describedabove are measured or calculated are often very difficult to obtain. Insuch cases, the processes described above can be used to determinerefractive index data from a measured extinction spectrum measured for asample colloid in which the particle size distribution is known andparticle concentrations are known. Once obtained, this refractive indexdata can then be used to compute the particle size distribution andparticle concentration for a sample of particles of the same particlematerial and whose particle size distribution is unknown, but theparticle sizes are within the range of particle sizes of the processdescribed herein.

As described above, particle size distributions and particleconcentrations can be calculated for a sample of small particles whenthe refractive indices at respective wavelengths within the measurementwavelength range of the particle material and the fluid of a samplecolloid are known and the measured extinction spectrum of the samplecolloid is provided. Conversely, if the measured extinction spectrum andthe particle size distribution in the sample colloid are known, therefractive index/indices over the measurement wavelength range of theparticle material(s) can be calculated to provide a refractive indexvector. However, this is also a difficult problem to solve, mainlybecause there are multiple solutions for the refractive index (RI)vector all but one of which are wrong. Below are three embodiments ofmethods for calculating refractive index according to the presentinvention.

Generally, when computing a refractive index vector (representingwavelength, real refractive index, and imaginary refractive index) ofthe particle material of a sample of small particles, a sample colloidis provided in which the particle size distribution of the smallparticles in the sample colloid is known. The sample colloid is a dilutecolloid. The real refractive index value is an indicator of how muchlight is scattered and the imaginary refractive index value is anindicator of how much light is absorbed and turned into heat. A measuredextinction spectrum of the sample colloid is obtained, based onspectrophotometric measurement of the small particles in the samplecolloid at discrete wavelengths. An objective function of refractiveindex values is_minimized as a function of wavelength. This is done insubstantially the same way as solving for the concentration vector, butwhere the refractive index is found that minimizes the error between themeasured extinction spectrum and that computed from the known particlesize distribution, using the same Nelder-Meade algorithm to determinethe refractive index vector of the particle material of the smallparticles.

Since it is often difficult to obtain a monodisperse sample of smallparticles for refractive index measurement, the present methods andsystem include the capability to enter a measured particle sizedistribution along with the measured extinction spectrum. When therefractive index of the particle material is to be measured, theparticle size distribution is measured using a method that does notrequire the refractive index of the particle material to be known.Typically the particle size distribution measurements are provided inthe form of histograms on a spread sheet or other flat file. However, aPSD can be provided in other formats, as all a computer needs is a tableof particle size and numerical concentration for each possible particlesize. Accordingly, the algorithms are programmed to read a text or .csvfile containing the data representing the particle size distribution asa set of (particle size, concentration number) vectors. After the dataare read, the particle concentrations are converted from a numberdensity (number of particles pre unit volume of colloid) to a particlevolume fraction. Particle volume fraction is related to particle numberdensity by the following equation:

$\begin{matrix}{C_{v} = {\frac{4}{3}\pi \; a^{3}N}} & (12)\end{matrix}$

where N is the number density, α is the particle radius, and C_(v) isthe calculated volume fraction. In this case, α is known. The totalnumber of particles measured in the sample is not the number density,but is assumed to be proportional to the number density N. However, theparticle size distribution information can be used to generate anormalized PSD by volume. Equation (12) is used to calculate a volumefraction for each particle size. That is, for each particle size, thevolume fraction for that particle size is calculated by multiplying thenumber density of particles of that particle size by the cube of theparticle radius of the particles of that particle size, and a constant,as noted above in equation (12). The calculated volume fractions,plotted against the particles sizes, gives a distribution proportionalto particle volume. The distribution is then normalized so that the sumof the particle volume fractions is 1.

The normalized PSD together with the total volume fraction of particlescan then be used to calculate refractive index. Calculation of theextinction spectrum of a sample dilute colloid having a monodisperseparticle size distribution has already been described above, includingcalculating AU(λ) as a function of wavelength including the forwardscattering component. For sample dilute colloids with a polydisperseparticle size distribution, linear methods of solving are also applied,as a summation over the particle size distribution is performed tocompute the spectrum. That is, the extinction as a function ofwavelength is computed as follows:

$\begin{matrix}{{{AU}\left( {\lambda,n_{p},n_{f}} \right)} = {C_{v}{\sum\limits_{n}\; {{{PSD}\left( a_{n} \right)}{{AU}\left( {\lambda,a_{n},n_{p},n_{f}} \right)}}}}} & (13)\end{matrix}$

where C_(v) is the total volume concentration, PSD(α_(n)) is thenormalized volume fraction at the nth particle size an in the particlesize distribution (as described in the previous section), andAU(λ,α_(n),n_(p),n_(f)) is the computed absorbance with unity volumeconcentration at wavelength λ and particle size α_(n). n_(p) and n_(f)are refractive indices of the particle material and the fluid,respectively. Equation (13) is used extensively in all of the refractiveindex fitting algorithms described below.

The objective or cost function that the algorithms minimize is:

|AU(λ,n_(p),n_(f))−V_(m)(λ)|²  (14)

where V_(m) is the measured extinction spectrum. That is, the algorithmsfunction to find n_(p) as a function of wavelength that minimizesequation (14). In the first and third methods/algorithms describedbelow, this minimization is performed at each wavelength in the measuredextinction spectrum, typically 92 wavelengths spaced by 10 nm. In thesecond method/algorithm described below (Sellmeier fit), the objectivefunction (14)) is calculated for each wavelength in the wavelength rangeof the measured extinction spectrum and summed over all the wavelengthsto give a measure of the overall fit quality.

One method of calculating refractive indices will be referred to as apoint-by-point refractive index solver. Using this method, the measuredextinction spectrum of the sample colloid, the PSD of the samplecolloid, the concentrations of the particles of each of the particlesizes in the sample colloid, the transparent wavelength range (thespectral region in which there is no significant absorption, that is,the imaginary refractive index is essentially zero, and therefore onlythe real refractive index is solved for), and an initial refractiveindex. Since the extinction coefficient measured by thespectrophotometer depends on both the refractive index and the particleconcentration, the method requires an input of the concentration vectorto calculate the refractive index. Alternatively, if the refractiveindex at one wavelength is known, this can be used to calculate theconcentration. This technique can therefore be used when theconcentration is not known or when a refractive index for the particlematerial at a particular wavelength is known. Often the refractive indexat the sodium d-line (589 nm) can be found in the literature. Theprogram then computes the real refractive index at successiveuser-specified wavelength intervals starting at the longest wavelengthand finishing at the absorption edge specified by the shortestwavelength in the transparency range. A refractive index maximum slopelimit prevents the algorithm from jumping discontinuously to anincorrect solution. The method works well for smaller particles, e.g.,particle sizes less than about 0.6 μm. For larger particles, the slopeconstraint is not sufficient to avoid wrong solutions.

To use the point-by-point refractive index solver, a user inputs (or,alternatively an automatic input can be performed of) the measuredextinction spectrum of the sample colloid, the cell length, the PSD(including the particle concentrations), the transparent wavelengthrange (which may be user specified), the wavelength interval between RIcalculations, and a starting RI at the longest wavelength in thetransparent wavelength range. The user also enters (or the system mayautomatically enter, upon receiving these as inputs) slope constraintsthat take the form of an allowed incremental change in the refractiveindex n for a wavelength step. The transparent wavelength range may notalways be known, but it can be discovered indirectly by checking to seeif the RI fits to different PSDs of the same particle material give thesame result. After solving for the refractive index as a function ofwavelength, a plot of the objective function (equation 14) can be madeagainst wavelength to easily identify where the real refractive indexsolution begins to fail. It is sometimes possible to detect the onset ofabsorption by looking at features in the extinction spectrum.

As noted previously, the point-by-point refractive index solver may havedifficulty finding the correct solution for a calculation of RI oflarger size particles. FIG. 11A is a plot 1100 of the measuredextinction spectrum of a sample colloid in which 1 μm polystyreneparticles prepared by Dow Chemical are dispersed at a concentration ofabout 0.1% by volume in water. When the objective function (equation(14)) is plotted as a function of wavelength and real particlerefractive index, the two-dimensional “roadmap” of potential refractiveindex solutions 1150 is plotted, as shown in FIG. 11B. The shadedregions 1153 correspond to an objective function magnitude <10⁻⁴, wherethe fit is good. Note that there are multiple solutions in regions wherethe wavelength becomes much shorter than the particle size. Thecross-over regions 1152, 1154 correspond to inflection point regions1102, 1104 on the spectrum curve 1100 of FIG. 11A. The problem with thepoint-by-point method is that it tends to take the lower path asindicated by arrow 1156 in the objective function plot, whereas, in thiscase, the correct solution is arrived at by taking the upper pathindicated by arrow 1158. FIG. 11C illustrates the result 1170 found bythe point-by-point method for this example, taking the pathway of arrow1156, with an allowed incremental RI change of +/−0.05 in a 10 nmwavelength increment.

Tightening the allowed incremental RI change (slope tolerance) to+0.02/−0.0001 RI in a 10 nm wavelength increment caused thepoint-by-point method to stay on the right path 1158, but the resultingRI curve 1180 was still too jagged especially near the inflectionpoints, as illustrated in FIG. 11D. For comparison purposes, toillustrate the jaggedness of the RI curve 1180 for 1 μm polystyreneparticles, the RI curve 1182 for 300 nm polystyrene particles that wascalculated by the point-by-point method under the same conditions isshown. Only results for wavelengths longer than 270 nm have beenplotted. It can be readily observed that curve 1182 is much smoother.The curve 1182 was found to work well for polystyrene samples, and was abetter RI estimate than the RI estimate provided by curve 1180 for the 1μm polystyrene particles.

Four different constrained optimization routines were used asalternatives for finding the refractive index vector as a function ofwavelength that minimizes the objective function (equation (14)) in theconstrained RI range. These were Nelder-Mead, Differential Evolution,Simulated Annealing, and Random Search, all of which are built intoMATHEMATICAL®. The Nelder-Mead routine, also known as the simplexmethod, was preferred because it is substantially faster than theothers. The second preferred routine was the Differential Evolutionroutine, which is described by the MATHEMATICAL® user guide ascomputationally expensive, but relatively robust and tends to work wellfor problems that have more local minima.

A second method of calculating refractive indices is referred to as theSellmeier real refractive index solver. Using this method, the realrefractive index is fit to a smooth curve. Unlike the point-by-pointmethod, the Sellmeier equation is based on a physical model fordispersion. It ensures that the solution is continuous and has normaldispersion (the refractive index decreases with increasing wavelength).It also suppresses measurement noise. This prevents the jagged errorsthat can occur when calculating refractive indices using larger-sizeparticles, and the point-by-point approach are avoided. The smooth curvethat the real refractive index is fitted to is derived from theSellmeier equation, which is often used to describe the refractive indexdispersion of optical materials. The Schott Glass catalog (Schott AG,Mainz, Germany) lists the six Sellmeier coefficients for an extensiveset of optical glasses and states that the refractive index is accurateto 5 parts per million. Excellent results were achieved by fitting aspectrum, calculated using the polystyrene RI determined using theSellmeier equation to the measured extinction spectrum.

The Sellmeier equation is an empirical relationship between refractiveindex and wavelength that is commonly used to describe dispersion ofoptical glasses. The usual form for glasses is:

$\begin{matrix}{{n^{2}(\lambda)} = {1 + \frac{B_{1}\lambda^{2}}{\lambda^{2} - C_{1}} + \frac{B_{2}\lambda^{2}}{\lambda^{2} - C_{2}} + \frac{B_{3}\lambda^{2}}{\lambda^{2} - C_{3}}}} & (15)\end{matrix}$

The optimization strategy for the Sellmeier fit involves finding the sixSellmeier coefficients that produce the best fit over the entirespectrum. Thus the objective function for use with the Sellmeier realrefractive index solver is defined as:

$\begin{matrix}{\sum\limits_{i}\; {{{{AU}\left( {\lambda_{i},n_{p},n_{f}} \right)} - {V_{m}\left( \lambda_{i} \right)}}}^{2}} & (16)\end{matrix}$

where summation is performed over all of the wavelengths in the measuredextinction spectrum V_(m). The Schott Glass catalog was referenced toselect the minimization constraints for the Sellmeier coefficients.These constraints can be as follows, but are subject to change as thismethod is further refined: 0.4<B1<1.5, 0.1<B2<0.8, 0.7<B3<2.0,0.001<C1<0.03, 0.005<C2<0.035, 80<C3<130. Using the same Nelder-Meadminimization algorithm used for the point-by-point refractive indexsolver described above, the refractive index result 1200 for 3 μmpolystyrene particles was produced as shown in FIG. 12A, using theSellmeier real refractive index solver.

The Sellmeier coefficients for this example were: {0.00187037,{B1→0.829063, B2→0.599588, B3→1.38423, C1→0.0118092, C2→0.0292444,C3→115.424}}; where the first number is the value of the objectivefunction (equation (16)) after fitting. The Sellmeier coefficients wereused to compute the particle refractive index using the point-by-pointobjective function (equation (14)) to get a sense of how well therefractive index fits across the spectrum. The result 1210 of thiscalculation for 3 μm polystyrene particles is shown in FIG. 12B. The fitmerit is computed by plotting the objective function (equation (14))using the computed refractive index η at each wavelength to provide ameasure of the fit quality at each point. A typical threshold fit valueis set a 10⁻⁴, wherein a fit merit value less than or equal to about10⁻⁴ indicates a good fit. It can be observed in FIG. 12B that the fitwas reasonably good, except in the UV wavelengths at the short end ofthe spectrum.

A third method of calculating refractive indices is referred to as thepoint-by-point complex refractive index solver. This method provides alogical extension of the point-by-point refractive index solver fromreal refractive indices to complex refractive indices, for use withopaque materials. In this approach, two spectral measurements arerequired to solve for the complex refractive index, since thecomputation of two variables is required. Thus, the extinction spectraof two dilute colloids with different particle sizes are measured inorder to compute the refractive index. The objective function that isminimized using this approach is sum of two functions of the form inequation (14), one for each spectrum.

FIGS. 13A and 13B show the real and imaginary refractive index plots1300 and 1350, respectively, for polystyrene particles. The firstmeasured spectrum was measured from a monodisperse colloid of 74 nmpolystyrene particles and the second measured spectrum was measured froma monodisperse colloid of 155 nm polystyrene particles. Thepoint-by-point refractive index solver method was used for the real partcalculations at wavelengths greater than about 270 nm. For shorterwavelengths, the point-by-point complex refractive index solver methodwas used, with a starting refractive index guess with k=0 and n=the 280nm real refractive index resulting from the real refractive indexsolution provided by the point-by-point refractive index solver methodThe term “k” is defined as the imaginary component of the refractiveindex, and “n” is the real component of the refractive index, which canbe written as ñ=n+ik, where i is the square root of −1. The tilde abovethe n on the left side of the equation means the n is a complex number,a fairly common notation. Starting at 270 nm, incremental changes in nof +/−0.05 and in k of +/−0.1 were allowed in each of the 5 nmwavelength increments. This method requires that the initial n and kassignments be positive numbers.

In examining the fit merit for these results, it was found that the fitwas good everywhere except near 230 nm. It was postulated that the badfit results near 230 nm may have been caused by some contamination inthe sample(s).

FIG. 14 illustrates a typical computer system 1400 in accordance with anembodiment of the present invention. The computer system 1400 may beincorporated into a spectrophotometer system, or may be configured toreceive spectrophotometric data from a spectrophotometer via interface1410, for example, and with user interaction via interface 1410 of thesystem 1400, such as in conjunction with user interface 260. Computersystem 1400 includes any number of processors 1402 (also referred to ascentral processing units, or CPUs) that are coupled to storage devicesincluding primary storage 1406 (typically a random access memory, orRAM), primary storage 1404 (typically a read only memory, or ROM).Primary storage 1404 acts to transfer data and instructionsuni-directionally to the CPU and primary storage 1406 is used typicallyto transfer data and instructions in a bi-directional manner. Both ofthese primary storage devices may include any suitable computer-readablemedia such as those described above. A mass storage device 1408 is alsocoupled bi-directionally to CPU 1402 and provides additional datastorage capacity and may include any of the computer-readable mediadescribed above. Mass storage device 1408 may be used to store programs,such as PSD calculation programs, refractive index calculation programs,post processing programs, and the like and is typically a secondarystorage medium such as a hard disk that is slower than primary storage.It will be appreciated that the information from primary storage 1406,may, in appropriate cases, be stored on mass storage device 1408 asvirtual memory to free up space on primary storage 1406, therebyincreasing the effective memory of primary storage 1406. A specific massstorage device such as a CD-ROM or DVD-ROM 1414 may also pass datauni-directionally to the CPU.

CPU 1402 is also coupled to an interface 1410 that includes one or moreinput/output devices such as video monitors, user interface, trackballs, mice, keyboards, microphones, touch-sensitive displays,transducer card readers, magnetic or paper tape readers, tablets,styluses, voice or handwriting recognizers, or other well-known inputdevices such as, of course, other computers. Finally, CPU 1402optionally may be coupled to a computer or telecommunications networkusing a network connection as shown generally at 1412. With such anetwork connection, it is contemplated that the CPU might receiveinformation from the network, or might output information to the networkin the course of performing the above-described methods. Theabove-described devices and materials are known in the computer hardwareand software arts.

The hardware elements described above may operate in response to theinstructions of multiple software modules for performing the operationsof this invention. For example, instructions for calculating particlesize distributions and concentrations and instructions for postprocessing may be stored on mass storage device 1408 or 1414 andexecuted on CPU 1408 in conjunction with primary memory 1406.

While the present invention has been described with reference to thespecific embodiments thereof, it should be understood that variouschanges may be made and equivalents may be substituted without departingfrom the scope of the invention.

1. A method for computing small particle size distributions, comprising:providing a reference matrix of pre-computed or pre-measured referencevectors, each reference vector representing a discrete particle size ofa particle size distribution of particles contained in a dilute colloid,each reference vector representing a reference extinction spectrum overa predetermined wavelength range; providing a measurement vectorrepresenting a measured extinction spectrum of the sample particles inthe sample colloid, wherein the measured extinction spectrum has beenspectrophotometrically measured; determining at least one of a particlesize, the particle size distribution and at least one particleconcentration of the particles in the sample dilute colloid, using thereference matrix, the measurement vector and linear equations; andsmoothing results calculated for the determining at least one of aparticle size, the particle size distribution and at least one particleconcentration of the particles in the sample dilute colloid.
 2. Themethod of claim 1, wherein each reference vector includes pairs ofwavelength and extinction values measured over predetermined wavelengthsspectrophotometrically.
 3. The method of claim 1, wherein saiddetermining comprises applying a non-negative least squares algorithm tothe reference matrix and the measurement vector to solve for theparticle size distribution and particle concentrations of the sampleparticles.
 4. The method of claim 1, wherein particle sizes of theparticles in the sample dilute colloid are in the range of about 10 nmto about 15 μm.
 5. The method of claim 1, wherein the referenceextinction spectra comprise extinction values determined from Miescattering spectra.
 6. The method of claim 1, wherein the wavelengthrange is from about 190 nm to about 1100 nm.
 7. The method of claim 1,wherein said determining comprises: calculating a least squares errorbetween the reference matrix and the measurement vector; and finding aconcentration vector that minimizes the least squares error, wherein theconcentration vector represents the particle concentrations of theparticles in the sample dilute colloid.
 8. The method of claim 1,wherein said determining includes iteratively removing from thereference matrix at least one of the reference vectors that represents alargest discrete particle size and each time determining the particlesize distribution and concentrations of particles in the sample dilutecolloid until an error in a fit of the measurement vector to thereference matrix that is greater than a predetermined error tolerance iscalculated.
 9. The method of claim 1, further comprising: compensatingfor forward scattering effects by applying a forward scatteringcorrection factor to values of the reference matrix.
 10. The method ofclaim 1, further comprising: outputting at least one of the at least oneof a particle size, the particle size distribution and at least oneparticle concentration of the particles in the sample dilute colloid foruse by a user.
 11. A method for computing at least one of small particlesize and small particle concentration, said method comprising: providinga reference matrix of pre-computed or pre-measured reference vectors,each reference vector representing a discrete particle size of aparticle size distribution of particles contained in a dilute colloid,each the reference vector including a reference extinction spectrum overa predetermined wavelength range; providing a measurement vectorrepresenting a measured extinction spectrum of the sample particles inthe sample colloid, wherein the measured extinction spectrum has beenspectrophotometrically measured; determining the particle sizedistribution in the sample dilute colloid, using linear equations, thereference matrix and the reference vector; determining whether thedetermined particle size distribution is a broad distribution, based ona predetermined percentage of non-zero particle size values in theparticle size distribution; when the determined particle sizedistribution is not a broad distribution, finding a best fit solutionfrom the determined particle size distribution and particleconcentrations relative to the matrix; and when the determined particlesize distribution is a broad distribution, iteratively removing from thereference matrix at least one of the pre-computed reference vectors thatrepresents a largest discrete particle size and each time determiningthe particle size distribution and particle concentrations in the sampledilute colloid until an error in a fit of the measurement vector to thereference matrix that is greater than a predetermined error tolerance iscalculated, and finding a best fit solution from the determined particlesize distribution and particle concentrations relative to the matrixafter the iteratively removing.
 12. The method of claim 11, wherein eachsaid reference vector includes pairs of wavelength and extinction valuesmeasured over discrete wavelengths spectrophotometrically.
 13. Themethod of claim 11, wherein said best fit solution when the determinedparticle size distribution is not a broad distribution is a solutionproduced by one of: smoothing the determined particle size distributionand particle concentrations, selecting a single match size from thedetermined particle size distribution that best matches the measurementvector, and selecting a single match size from the determined particlesize distribution that best matches the measurement vector and smoothingthe best single match.
 14. The method of claim 13, wherein saidsmoothing comprises Gaussian blurring.
 15. The method of claim 11,further comprising smoothing said best fit solution when the determinedparticle size distribution is a broad distribution.
 16. The method ofclaim 15, wherein said smoothing comprises Gaussian blurring.
 17. Themethod of claim 11, wherein the provision of the measurement vectorcomprises: measuring by spectrophotometry the measurement vector,wherein the extinction values are measured at discrete wavelengths. 18.A system for measuring at least one of small particle size and smallparticle concentration, said system comprising: a processor; a referencematrix of pre-computed or pre-measured reference vectors stored inmemory that is accessible by the processor, each reference vectorrepresenting a discrete particle size of a particle size distribution ofparticles contained in a dilute colloid, each the reference vectorincluding a reference extinction spectrum over a predeterminedwavelength range; and programming configured to be run by the processor,when the system is provided with a measurement vector representing ameasured extinction spectrum of the sample particles in the samplecolloid, wherein the measured extinction spectrum has beenspectrophotometrically measured, to cause the processor to performoperations comprising: determining at least one of a particle size, theparticle size distribution, and at least one particle concentration ofthe particles in the sample colloid, using linear equations, thereference matrix and the reference vector; and smoothing resultscalculated for the determining at least one of a particle size, theparticle size distribution and at least one particle concentration ofthe particles in the sample dilute colloid.
 19. The system of claim 18,wherein each said reference vector includes pairs of wavelength andscattering values measured over discrete wavelengthsspectrophotometrically.
 20. The system of claim 18, further comprisingprogramming configured to be run by the processor to pre-compute thereference matrix of pre-computed reference vectors.
 21. The system ofclaim 18, further comprising: a spectrophotometer configured to measurethe extinction values of the sample particles in the sample colloid, thespectrophotometer further being configured to perform at least one ofcalculating and inputting the measurement vector of the extinctionvalues, and inputting the extinction values of the sample particles inthe sample colloid to the processor.
 22. The system of claim 18, whereinsaid programming is configured to determine the particle sizedistribution and particle concentrations in the sample dilute colloid byapplying a non-negative least squares algorithm to solve for theparticle size distribution and particle concentrations in the sampledilute colloid, using the reference matrix and the measurement vector asinputs.
 23. The system of claim 18, wherein said discrete wavelengthsare within a range of wavelengths from about 190 nm to about 1100 nm.24. The system of claim 18, wherein said processor executes theprogramming to calculate a least squares error between the referencematrix and the measurement vector, and find a concentration vector thatminimizes the least squares error.
 25. The system of claim 18, whereinsaid processor executes programming to compensate for forward scatteringeffects by applying a forward scattering correction factor to values ofthe reference matrix.
 26. A computer readable medium carrying one ormore sequences of instructions for computing at least one of smallparticle size and small particle concentration is provided, whereinexecution of the one or more sequences of instructions by one or moreprocessors causes the one or more processors to perform a processcomprising: receiving a reference matrix of pre-computed or pre-measuredreference vectors, each reference vector representing a discreteparticle size of a particle size distribution of particles in a dilutecolloid, each the reference vector representing a reference extinctionspectrum over a predetermined wavelength range; receiving a measurementvector representing a measured extinction spectrum of the sampleparticles in the sample colloid, wherein the measured extinctionspectrum has been spectrophotometrically measured; determining at leastone of a particle size, the particle size distribution and at least oneparticle concentration of the particles in the sample dilute colloid,using the reference matrix, the measurement vector and linear equations;and smoothing results calculated for the determining at least one of aparticle size, the particle size distribution and at least one particleconcentration of the particles in the sample dilute colloid.
 27. Thecomputer readable medium of claim 26, wherein each said reference vectorincludes pairs of wavelength and extinction values measured overdiscrete wavelengths spectrophotometrically.
 28. A method for computinga refractive index of small particles, said method comprising: providinga particle size distribution of the small particles in a sample colloid;computing an extinction spectrum of the small particles in the samplecolloid, based on spectrophotometric measurement of the small particlesin the sample colloid at discrete wavelengths; and minimizing anobjective function of refractive index values as a function ofwavelength to determine the refractive index of the small particles. 29.The method of claim 28, further comprising: plotting a fit merit plot toshow fit quality of the determined refractive index versus wavelength.30. The method of claim 28, wherein the refractive index is determinedusing a point-by-point real refractive index solver application.
 31. Themethod of claim 28, wherein the refractive index is determined using aSellmeier real refractive index solver application.
 32. The method ofclaim 28, wherein the refractive index is determined using apoint-by-point complex refractive index solver application.